A Hysteresis Approach to Phase--eld Models

Phase-eld systems as mathematical models to forecast the evolution of processes involving phase transitions have drawn a considerable interest in recent years. However, while they are capable of capturing many of the experimentally observed phenomena, they are only of restricted value in modelling hysteresis eeects occuring during phase transition processes. To overcome this shortcoming, a new approach to phase-eld models is proposed in this paper which is based on the mathematical theory of hysteresis operators developed in the past fteen years. The approach taken here leads to highly nonlinearly coupled systems of diierential equations containing hysteretic nonlinearities at diierent places. For such a system, well-posedness and thermodynamic consistency are proved. Due to the lack of smoothness (hysteresis operators are, as a rule, non-diierentiable) in the system, the method of proof has to be diierent from those usually employed for classical phase-eld systems.